Abstract:

In the context of forest products, a cutting order is a list of dimension parts along
with demanded quantities. The cutting-order problem is to minimize the total cost of
filling the cutting order from a given lumber grade (or grades). Lumber of a given grade
is supplied to the production line in a random sequence, and each board is cut in a way
that maximizes the total value of dimension parts produced, based on a value (or price)
specified for each dimension part. Hence, the problem boils down to specifying suitable
dimension-part prices for each board to be cut.
The method we propose is adapted from Gilmore and Gomory's linear programming
approach to the cutting stock problem. The main differences are the use of a random
sample to construct the linear program and the use of prices rather than cutting patterns
to specify a solution. The primary result of this thesis is that the expected cost of
filling an order under the proposed method is approximately equal to the minimum possible
expected cost, in the sense that the ratio (expected cost divided by the minimum
expected cost) approaches one as the size of the order (e.g., in board feet) and the size of
the random sample grow large.
A secondary result is a lower bound on the minimum possible expected cost. The
actual minimum is usually impractical to calculate, but the lower bound can be used in
computer simulations to provide an absolute standard against which to compare costs. It
applies only to independent sequences, whereas the convergence property above applies
to a large class of dependent sequences, called alpha-mixing sequences.
Experimental results (in the form of computer simulations) suggest that the proposed
method is capable of attaining nearly minimal expected costs in moderately large
orders. The main drawbacks are that the method is computationally expensive and of
questionable value in smaller orders.